Laplacian quantum walks on blow-up graphs

This paper is a sequel to the work of Bhattacharjya et al. (2024) [2] on quantum state transfer on blow-up graphs, where instead of the adjacency matrix, we take the Laplacian matrix as the time-independent Hamiltonian associated with a blow-up graph. We characterize Laplacian strong cospectrality, periodicity, perfect state transfer (LPST) and pretty good state transfer (LPGST) on blow-up graphs. We present several constructions of blow-up graphs with LPST and produce new infinite families of regular graphs where each vertex is involved in LPST. We also determine LPST and LPGST in blow-ups of classes of trees. Finally, if n ≡ 0 (mod 4), then the blow-up of n copies of a graph has no LPST, but we show that under certain conditions, the addition of an appropriate matching to this blow-up graph results in LPST.